Astronomy Principles and Practice Fourth Edition.pdf

Spectrometry 325
produced and the ability to record fine detail within the spectrum. The first may be described by the
term angular dispersion and the second by the term spectral resolving power.
The angular dispersion, AD, of any spectrometer is defined as the rate of change of the dispersed
collimated beams with wavelength. Hence,
AD = dθ
dλ .
This may be converted to a linear dispersion, LD, i.e. the length of spectrum produced representing
a certain wavelength interval according to the focal length, F, of the camera lens (see equation 16.2).
Hence,
LD = dθ F. (19.14)
dλ
It is usual practice to express the relationship between the wavelength range covered in the spectral
plane by the reciprocal linear dispersion (RLD), usually in units of Åmm −1 .
The expression for the angular dispersion of a prism is complicated and will not be given here.
It depends on the angle of the prism and the refractive index of the glass. As the second parameter
is wavelength-dependent—the very property which is being made use of to produce the spectrum—
the angular dispersion and, hence, the reciprocal linear dispersion will vary with wavelength. Equal
wavelength intervals are, therefore, not covered by equal distances within the spectrum.
In the case of the diffraction grating, the angular dispersion may be obtained by differentiating
equation (19.13). Thus,
dθ
dλ = m
d cos θ .
By designing a grating spectrometer so that θ is close to zero, cos θ will not differ significantly
from unity and the angular dispersion will be practically constant for all wavelengths. The spectrum
produced will have a scale which is linear, this being just one of the advantages that the grating has
over the prism.
The spectral resolving power of any spectrometer is defined as its ability to allow inspection of
elements of a spectrum which are close together. If a part of the spectrum centred at a wavelength, λ,
is being investigated and λ + λis the closest wavelength to λ which can be seen distinctly as being
separate from λ, then the spectral resolving power, R, isdefinedas
R =
λ
λ . (19.15)
The largest absorption features in a stellar spectrum are the order of 10 Å wide and it will be
seen from equation (19.15) that a resolving power greater than 10 3 would be required to record them.
A resolving power greater than 10 4 is needed to investigate spectral details which cover a range of
1 Å or less. If stellar absorption line profiles are to be recorded to real purpose, the resolving power
must, in general, be at least of the order of 10 5 . It may also be pointed out that the reciprocal
of the spectrometer’s resolving power is equal to the expression describing the Doppler shift (see
equation (15.25)) and, hence, the value of R expresses the ability of a spectrometer to detect such
shifts. Unless the Doppler shift is larger than a resolved spectral element, it obviously would not be
detected.
There is a theoretical limit to the spectral resolving power of any spectrometer. Assuming that
the aberrations of the optical elements within the spectrometer in no way cause deterioration in the
instrument’s performance, the resolving power is limited by the size of the dispersing element—prism
or diffraction grating.
For a prism, the theoretical resolving power is given by
R = t dn
(19.16)
dλ